NDSolve
NDSolveValue
DirichletCondition

Finite Element Method

Transient Gray–Scott Model

Animate a solution to reaction diffusion equations over a disk

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Simulate a Gray–Scott model that leads to interesting patterns over a disk. The Gray–Scott equations are a reaction diffusion system of coupled nonlinear transient equations used in, among other areas, biology.

∂u

∂t

+∇(-

∇u)+(

+f)u

∂v

∂t

+∇(-

∇u)+(-uv+f+k)v

Set up the equations and specify coefficients:

In[1]:=

eqn=

(1,0,0)

[t,x,y]+

∇

{x,y}

-c1

∇

{x,y}

u[t,x,y]+(v[t,x,y]^2+f)u[t,x,y]f,

(1,0,0)

[t,x,y]+

∇

{x,y}

-c2

∇

{x,y}

v[t,x,y]+(-u[t,x,y]v[t,x,y]+f+k)v[t,x,y]0//.{c12*10^-5,c2c1/4,f1/25,k3/50};

As initial conditions, set

u=1/2

and

v=1

≤1/40

, else 0:

In[2]:=

ics={u[0,x,y]1/2,v[0,x,y]If[x^2+y^2≤1/40,1,0]};

Set both

and

to zero on the boundary:

In[3]:=

bcs=DirichletCondition[{u[t,x,y]0,v[t,x,y]0},True];

Solve the system of equations on a mesh with specified spacing:

In[4]:=

{ufun,vfun}=NDSolveValue[{eqn,bcs,ics},{u,v},{x,y}∈Disk[],{t,0,3000},Method{"PDEDiscretization"{"MethodOfLines","SpatialDiscretization"{"FiniteElement","MeshOptions"{"MaxCellMeasure"0.003}}}}]

Out[4]=



InterpolatingFunction



Domain: 0.,3.00×

,{-1.,1.},{-1.,1.}

Output: scalar

Data not saved. Save now



InterpolatingFunction



Domain: 0.,3.00×

,{-1.,1.},{-1.,1.}

Output: scalar

Data not saved. Save now





Visualize the result:

In[5]:=

{vmin,vmax}=MinMax[vfun["ValuesOnGrid"]];frames=TableContourPlotvfun[t,x,y],{x,-1,1},{y,-1,1},

,{t,100,2000,25};ListAnimate[frames]

Out[5]=

Related Symbols

NDSolve
NDSolveValue
DirichletCondition

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